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Eigenvalues of quasibounded maximal monotone operators
Journal of Inequalities and Applications volume 2014, Article number: 21 (2014)
Abstract
Let X be a real reflexive separable Banach space with dual space and let L be a dense subspace of X. We study a nonlinear eigenvalue problem of the type
where is a strongly quasibounded maximal monotone operator and satisfies the condition with . The method of approach is to use a topological degree theory for -perturbations of strongly quasibounded maximal monotone operators, recently developed by Kartsatos and Quarcoo. Moreover, applying degree theory, a variant of the Fredholm alternative on the surjectivity of the operator is discussed, where we assume that λ is not an eigenvalue for the pair , T and C are positively homogeneous, and C satisfies the condition .
1 Introduction and preliminaries
A systematic theory of compact operators emerged from the theory of integral equations of the form
Here, is a parameter, y and k are given functions, and x is the unknown function. Such equations play a role in the theory of differential equations. The study goes back to Krasnosel’skii [1]. Moreover, the eigenvalue problem of the form
could be solved with the Galerkin method, where C is continuous, bounded, and of type ; see, e.g., [2].
From now on, we concentrate on the class of maximal monotone operators, as a generalization of linear self-adjoint operators. The theory of nonlinear maximal monotone operators started with a pioneer work of Minty [3] and has been extensively developed, with applications to evolution equations and to variational inequalities of elliptic and parabolic type; see [4, 5]. The eigenvalue problem for various types of nonlinear operators was investigated in [6–10]. As a key tool, topological degree theory was made frequent use of; for instance, the Leray-Schauder degree and the Kartsatos-Skrypnik degree; see [11–15].
Let X be a real reflexive Banach space with dual space . We consider a nonlinear eigenvalue problem of the form
where is a maximal monotone multi-valued operator and is a single-valued operator. In the case where the operator C or the resolvents of T are compact, it was studied in [6, 7, 10] by using the Leray-Schauder degree for compact operators. When the operator C is densely defined and quasibounded and satisfies the condition , Kartsatos and Skrypnik [9] solved the above problem (E) via the topological degree for these operators given in [13].
We are now focused on the quasiboundedness of the operator T instead of that of the operator C. Actually, a strongly quasibounded operator due to Browder and Hess [16] may not necessarily be bounded. One more thing to be considered is the condition , where L is a dense subspace of X with . In fact, the condition was first introduced in [12] and the structure of the class or was discussed in [17], as a natural extension of the class ; see [4, 14].
In the present paper, the first goal is to study the above eigenvalue problem (E) for strongly quasibounded maximal monotone operators, provided that the operator C satisfies the condition . In addition, we assume the following property : For , there exists a such that the inclusion
has no solution in , where Ω is a bounded open set in X and J is a normalized duality operator. This property is closely related to the use of a topological tool for finding the eigensolution on the boundary of Ω; see [9, 10]. To solve the above problem (E), we thus use the degree theory for densely defined -perturbations of maximal monotone operators introduced by Kartsatos and Quarcoo in [18]. Roughly speaking, the degree function is based on the Kartsatos-Skrypnik degree [8] of the densely defined operators , which is constant for all small values of t, where is the approximant introduced by Brézis et al. [19]. Such an approach was first used by Browder in [11]. The second goal is to establish a variant of a Fredholm alternative result on the surjectivity for the operator , where is not an eigenvalue for the pair and the operator C satisfies the condition ; see [9, 20].
This paper is organized as follows: In Section 2, we give some eigenvalue results for strongly quasibounded maximal monotone operators by applying the Kartsatos-Quarcoo degree theory. Section 3 contains a version of the Fredholm alternative for positively homogeneous operators, with a regularization method by means of a duality operator .
Let X be a real Banach space, its dual space with the usual dual pairing , and Ω a nonempty subset of X. Let , intΩ, and ∂ Ω denote the closure, the interior, and the boundary of Ω in X, respectively. The symbol → (⇀) stands for strong (weak) convergence. An operator is said to be bounded if A maps bounded subsets of Ω into bounded subsets of . A is said to be demicontinuous if, for every and for every sequence in Ω with , we have .
An operator is said to be monotone if
where denotes the effective domain of T.
The operator T is said to be maximal monotone if it is monotone and it follows from and
that and .
An operator is said to be strongly quasibounded if for every there exists a constant such that for all with
where , we have .
We say that satisfies the condition on a set if for every sequence in M with and every sequence with where , we have .
We say that satisfies the condition on a set if for every sequence in M with
we have .
Throughout this paper, X will always be an infinite-dimensional real reflexive separable Banach space which has been renormed so that X and its dual are locally uniformly convex.
An operator is said to be a duality operator if
where is continuous, strictly increasing, and as . When φ is the identity map I, is called a normalized duality operator.
It is described in [21] that is continuous, bounded, surjective, strictly monotone, maximal monotone, and that it satisfies the condition on X.
The following properties as regards maximal monotone operators will often be used, taken from [[19], Lemma 1.3], [[13], Lemma 3.1], [[22], Lemma 1], and [[18], Lemma D] in this order.
Lemma 1.1 Let be a maximal monotone operator. Then the following statements hold:
-
(a)
For each , the operator is bounded, demicontinuous, and maximal monotone.
-
(b)
If, in addition, and , then the operator , is continuous on .
Lemma 1.2 Let and be two maximal monotone operators with and such that is maximal monotone. Assume that there is a sequence in with and a sequence in such that and , where . Then the following hold:
-
(a)
The inequality is true.
-
(b)
If , then and .
Lemma 1.3 Let be a strongly quasibounded maximal monotone operator such that and . If is a sequence in and is a sequence in X such that
where S, are positive constants, then the sequence is bounded in .
Let L be a dense subspace of X and let denote the class of all finite-dimensional subspaces of L. Let be a sequence in the class such that for each
Set .
Definition 1.4 Let be a single-valued operator with . We say that C satisfies the condition if for every sequence in satisfying equation (1.1) and for every sequence in L with
for every , we have , and .
We say that C satisfies the condition if the operator , defined by , satisfies the condition for every .
We say that the operator C satisfies the condition if it satisfies the condition with ‘’ replaced by ‘’. We say that C satisfies the condition if the operator satisfies the condition for every .
It is obvious from Definition 1.4 that if the operator C satisfies the condition , then C satisfies the condition . However, the converse is not true in general, as we see in Example 3.2 of [17].
2 The existence of eigenvalues
In this section, we deal with some eigenvalue results for strongly quasibounded maximal monotone operators in reflexive separable Banach spaces, based on a topological degree theory for -perturbations of maximal monotone operators due to Kartsatos and Quarcoo [18].
We establish the existence of an eigenvalue concerning -perturbations of strongly quasibounded maximal monotone operators.
Theorem 2.1 Let Ω be a bounded open set in X with and let L be a dense subspace of X. Suppose that is a multi-valued operator and is a single-valued operator with such that
(t1) T is maximal monotone and strongly quasibounded with and ,
(c1) C satisfies the condition ,
(c2) for every and , the function , defined by , is continuous on F, and
(c3) there exists a nondecreasing function such that
Let Λ and be two given positive numbers.
-
(a)
For a given , assume the following property :
There exists a such that the inclusion
has no solution in .
Then there exists a such that
Here, denotes the intersection of and .
-
(b)
If property is fulfilled for every , T satisfies the condition on , , and the set is bounded, then the inclusion
has a solution in .
Proof (a) Assume that the conclusion of (a) is not true. Then for every , the following boundary condition holds:
Considering a multi-valued map H given by
the inclusion has no solution x in for all . Actually, this holds for , in view of the injectivity of the operator with .
Now we consider a single-valued map given by
We will first show that there exists a positive number such that the equation
has no solution x in for all and all . For , assertion (2.2) is obvious because implies . Assume that assertion (2.2) does not hold for any . Then there exist sequences in , in , and in such that , , , , and
where , , and . Let S be a positive upper bound for the bounded sequence . Note that . Indeed, if , then we have by the monotonicity of with , equation (2.3), and (c3)
and so ; but , which is a contradiction. Since we have the inequality
Lemma 1.3 implies that the sequence is bounded in the reflexive Banach space . Passing to a subsequence, if necessary, we may suppose that for some . Set
By equation (2.3), we have and hence
Recall that if two operators and are maximal monotone and , then the sum is also maximal monotone; see [[5], Theorem 32.I]. Since is thus maximal monotone and , Lemma 1.2(a) says that
From equations (2.3), (2.5), and the equality
it follows that
Since the operator C satisfies the condition , we find from equations (2.4) and (2.6) that and . Since , Lemma 1.2(b) tells us that and . From , we get
which contradicts our boundary condition equation (2.1). Consequently, we have proven our first assertion: that there exists a number such that
In the next step, we want to show that for each fixed , the degree is independent of , where d denotes the Kartsatos-Skrypnik degree from [12]. Fix . For , let be defined by
where for and for . First of all, for every finite-dimensional space and every , the function , defined by , is continuous on because the operators and J are continuous and C satisfies the condition (c2). To show that the family satisfies the condition , we assume that is a sequence in and is a sequence in such that , , and
for every , where and . By Lemma 1.1(a), the sequence is bounded in . So we may suppose without loss of generality that and for some . There are two cases to consider. If , then we have
which implies along with equation (2.7)
where S is an upper bound for the sequence . Hence it follows that , , and . Now let . We may suppose that for all . Set and . The relation (2.7) can be expressed in the form
From the second part of equation (2.8), it is obvious that
By the monotonicity of the operator , we have
Hence it follows from the first part of equation (2.8) and from equation (2.10) that
Since the operator C satisfies the condition , we find from equations (2.9) and (2.11) that
By the demicontinuity of the operators and J, we have
and hence
Consequently, the family satisfies the condition , as required.
Since for all , we see, in view of Theorem A of [18], that the degree is independent of the choice of . Until now, we have shown that for each fixed , the degree is constant for all . Notice that is maximal monotone and strongly quasibounded, , and
Combining this with our first assertion above, Theorem 2 of [18] says that for each fixed , the degree is constant for all . If deg denotes the degree introduced in [18], then for every , we have
and hence
where the last equality follows from Theorem 3 in [23]. Thus, for all , the inclusion
has a solution in , which contradicts property . We conclude that statement (a) is true.
-
(b)
Let be a sequence in such that . According to statement (a), there exists a sequence in such that
where . If we set , , and , it can be rewritten in the form
Notice that the sequence is bounded in . This follows from the strong quasiboundedness of the operator T together with the inequality
where S is an upper bound for the sequence . From equation (2.12), is bounded in . Without loss of generality, we may suppose that
where , , and . Note that the limit belongs to . In fact, if , then the boundedness of the set implies that and so by equation (2.12) . Since the maximal monotone operator T satisfies the condition on , we find from equation (2.13) and Lemma 1.2(b) that , , and , which contradicts the hypothesis that . As , we have
From equation (2.12) it follows that
where the last inequality follows from Lemma 1.2(a). Since the operator C satisfies the condition , we obtain from equations (2.14) and (2.15) and . By the maximal monotonicity of the operator T, we have and . We conclude that
This completes the proof. □
Remark 2.2 (a) In Theorem 2.1, it is inevitable that the set is assumed to be bounded because it does not hold in general that if then .
-
(b)
When C is quasibounded and satisfies the condition , it was studied in [[9], Theorem 4] by using Kartsatos-Skrypnik degree theory for -perturbations of maximal monotone operators developed in [13]. For the case where C is generalized pseudomonotone in place of the condition , we refer to [[20], Theorem 2.1].
From Theorem 2.1, we get the following eigenvalue result in the case when the operator C satisfies the condition .
Corollary 2.3 Let T, Ω, L, Λ, be as in Theorem 2.1. Suppose that is a strongly quasibounded demicontinuous operator such that
(c1′) C satisfies the condition on X,
(c2) for every and , the function , defined by , is continuous on F, and
(c3) there exists a nondecreasing function such that
Then the following statements hold:
-
(a)
If property is fulfilled for a given , then there exists a such that .
-
(b)
If property is fulfilled for every , T satisfies the condition on and , then the inclusion has a solution in .
Proof Statement (a) follows immediately from Theorem 2.1 if we only show that the operator C satisfies the condition with . To do this, let be given and suppose that is any sequence in X such that
for every . Then is obviously bounded from above. By the strong quasiboundedness of the operator C, the sequence is bounded in . Since is dense in the reflexive Banach space X, it follows from the third one of equation (2.16) that . Hence we obtain from the first and second one of equation (2.16)
Since C satisfies the condition on X and is demicontinuous, we have
Thus, the operator C satisfies the condition with .
-
(b)
Let be a sequence in such that . In view of (a), there exists a sequence in such that
(2.17)
where . Notice that the sequence is bounded in and so is . This follows from the strong quasiboundedness of the operator C and the inequality
We may suppose that , , and , where , , and . Note that belongs to . Indeed, if , then we have by the boundedness of and equation (2.17) and hence by the condition and , which contradicts the hypothesis . The rest of the proof proceeds analogously as in the proof of Theorem 2.1. □
Remark 2.4 (a) The boundedness assumption on the set is unnecessary in Corollary 2.3, provided that the operator C is strongly quasibounded.
-
(b)
An analogous result to Corollary 2.3 can be found in [[9], Corollary 1], where the operator C is supposed to be bounded.
We close this section by exhibiting a simple example of operators A satisfying the condition .
Let G be a bounded open set in . Let and . Define the two operators by
Then the operator is clearly bounded and continuous, and it satisfies the condition on X. The operator is compact; see [[24], Theorem 2.2] and [[5], Proposition 26.10]. In particular, the sum satisfies the condition with .
3 Fredholm alternative
In this section, we present a variant of the Fredholm alternative for strongly quasibounded maximal monotone operators, by applying Kartsatos-Quarcoo degree theory as in Section 2.
Given , an operator is said to be positively homogeneous of degree γ on a set if for all and all . For example, the duality operator is positively homogeneous of degree γ on X if for . In addition, the operators and given at the end of Section 2 are positively homogeneous of degree on .
Theorem 3.1 Let L be a dense subspace of X and let be given. Suppose that is an operator and is an operator with and such that
(t1) T is maximal monotone and strongly quasibounded with ,
(t2) implies for every , where ,
(c1) C satisfies the condition ,
(c2) for every and , the function , defined by , is continuous on F, and
(c3) there exists a nondecreasing function such that
If the operators T and C are positively homogeneous of degree γ on L, then the operator is surjective.
Proof Let be an arbitrary but fixed element of . For each fixed , consider a family of operators , given by
where for and for . The first aim is to prove that the set of all solutions of the equation is bounded, independent of . If , then implies . It suffices to show that is bounded. Assume the contrary; then there exist sequences in and in L such that , , and
which can be written as
We may suppose that for all . Since the operators T, C, and are positively homogeneous of degree γ, it follows from equation (3.1) that
Setting and , we have , , and
Then we obtain from equation (3.2) and (c3)
Hence the strong quasiboundedness of T implies that the sequence is bounded in . There are two cases to consider. If , then , , and the monotonicity of T with implies
which is a contradiction. Now let and set . Without loss of generality, we may suppose that
where , , and . By equation (3.2), we have and hence
Since the operator is maximal monotone, we have
In fact, if equation (3.4) is false, then there is a subsequence of , denoted again by , such that
Hence it is clear that
For every , we have, by the monotonicity of the operator ,
which implies along with equation (3.5)
By the maximal monotonicity of , we have and . Letting in equation (3.6), we get a contradiction. Thus, equation (3.4) is true.
Furthermore, equation (3.4) implies, because of , that
From equations (3.2), (3.7), and the equality
it follows that
Since the operator C satisfies the condition , we obtain from equations (3.3) and (3.8)
Since T is maximal monotone and is continuous, Lemma 1.2(b) implies that
Therefore, we obtain
which contradicts hypothesis (t2) with . Thus, we have shown that is bounded.
So we can choose an open ball in X of radius centered at the origin 0 so that
This means that for all . Note that the operator is maximal monotone, strongly quasibounded, , and the operator satisfies the condition and other conditions with for and for , where . Moreover, we know from Section 1 that the operator is continuous, bounded and strictly monotone, and that it satisfies the condition , and for .
Using the homotopy invariance property of the degree stated in [[18], Theorem 3], we have
Applying equation (3.9) with , there exists a sequence in L such that
Next, we show that the sequence is bounded in X. Indeed, assume on the contrary that there is a subsequence of , denoted by , such that . Dividing both sides of equation (3.10) by and setting and , we get
and so . Since for all , it follows from (t1) that the sequence is bounded in . We may suppose that and for some and some . As in the proof of equations (3.3) and (3.8) above, we can show that
for every . Since the operator C satisfies the condition , we obtain
By Lemma 1.2(b), we have and and hence
which contradicts hypothesis (t2) with . Therefore, the sequence is bounded in X.
Combining this with equation (3.10), we know from (c3) and (t1) that the sequence is also bounded in . Thus we may suppose that and for some and some . From and the maximal monotonicity of the operator T, we get as before
for every . Since the operator C satisfies the condition and T is maximal monotone, we conclude that
As was arbitrary, this says that the operator is surjective. This completes the proof. □
Remark 3.2 An analogous result to Theorem 3.1 was investigated in [[20], Theorem 4.1], where the method was to use Kartsatos-Skrypnik degree theory for quasibounded densely defined -perturbations of maximal monotone operators, developed in [13]; see also [[9], Theorem 5].
As a particular case of Theorem 3.1, we have another surjectivity result.
Corollary 3.3 Let L, T, and C be the same as in Theorem 3.1, except that hypothesis (t2) is replaced by
(t2′) for all .
If λ is not an eigenvalue for the pair , that is, implies , then the operator is surjective.
Proof Noting that
for every and , it is clear that hypothesis (t2) in Theorem 3.1 is satisfied. Apply Theorem 3.1. □
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This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012-0008345).
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KI conceived of the study and drafted the manuscript. BI participated in coordination. All authors approved the final manuscript.
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Kim, IS., Bae, JH. Eigenvalues of quasibounded maximal monotone operators. J Inequal Appl 2014, 21 (2014). https://doi.org/10.1186/1029-242X-2014-21
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DOI: https://doi.org/10.1186/1029-242X-2014-21